New Developments in Four Dimensions

该奖项为参加于2022年6月13日至17日在加拿大不列颠哥伦比亚省维多利亚大学举行的“四维新发展”会议的美国与会者提供旅费。会议将集中讨论与理解四维流形有关的低维拓扑分支。低维拓扑是对二维、三维和四维空间（流形）的数学研究。二维和三维空间是直观的：桌子、百吉饼或地球的表面都是二维空间的例子，我们居住的空间世界和百吉饼的内部都是三维空间的例子。四维空间更难想象（和研究），最突出的例子是宇宙的时空概念化。从这个意义上说，四维拓扑学是对我们的物理宇宙可能实现的可能形状的研究。表面在经典上已经得到了很好的研究，20世纪末和21世纪初的重大进展使研究人员对三维流形有了清晰的认识。四维流形代表了低维拓扑学的未知前沿，在这一数学领域仍有无数的开放猜想和未解之谜。在过去的几年中，在四维拓扑学中出现了爆炸性的活动，特别是在四流形的微分同构群的研究，外来结构和嵌入的构建和检测，以及作为该领域新工具的三分法的引入。本次会议的目的是聚集一个国际专家小组，探讨四维流形研究的最新发展，传播该领域的当代研究，促进在该领域工作的早期职业数学家的研究，包括和促进来自代表性不足群体的数学家的研究，并为新的合作的发生和旧的合作提供一个场所。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports travel for US-based participants in the conference “New Developments in Four Dimensions” held at the University of Victoria, British Columbia, Canada, during June 13-17, 2022. The meeting will focus on the branch of low-dimensional topology concerned with understanding four-dimensional manifolds. Low-dimensional topology is the mathematical study of spaces (manifolds) of dimension two, three, and four. Two-dimensional and three-dimensional spaces are intuitive: The surfaces of tables, bagels, or the earth are all examples of two-dimensional spaces and the spatial world we inhabit and the inside of a bagel are examples of three-dimensional spaces. Four-dimensional spaces are considerably more difficult to imagine (and study) with the most salient example being the spacetime conceptualization of the universe. In this sense, four-dimensional topology is the study of the possible shapes that our physical universe might realize. Surfaces have been well-studied classically, and major advances in the late 1900s and early 2000s have given researchers a clear understanding of three-dimensional manifolds. Four-dimensional manifolds represent the unknown frontier of low-dimensional topology, and there remain myriad open conjectures and unanswered question in this field of mathematics.There has been an explosion of activity within four-dimensional topology in the last few years, particularly in the study of diffeomorphism groups of four-manifolds, the construction and detection of exotic structures and embeddings, and the introduction of trisections as a new tool in the field. The purpose of this conference is to gather an international group of experts to explore these recent developments in the study of four-dimensional manifolds, disseminate contemporary research in the field, promote the research of early-career mathematicians working in the field, include and promote the research of mathematicians from underrepresented groups, and to give a venue for new collaborations to occur and old collaborations to continue.Conference Website: https://math.stanford.edu/~maggiehm/developmentsin4DThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.